Download Vocabulary: Shapes and Designs

Vocabulary: Shapes and Designs
Concept
Polygon: From examples of polygons and nonpolygons students create the definition of a
polygon as a closed figure made of straight,
non-intersecting edges. The focus is not on
learning names for polygons (triangle,
quadrilateral, pentagon, hexagon, septagon,
octagon etc.), but on reasoning about these
figures.
Example
A
E
Sorting polygons by properties: One can sort
B
polygons into regular or non-regular polygons; or
D
into concave and convex. (There are other ways
to sort polygons.) A regular polygon has sides
which are all equal and angle measures which are
C
all equal. A convex polygon “bulges” out; all
angles will be less than 180 degrees. A concave
polygon collapses in; at least one angle will be
“A” is a non-regular pentagon (5 sides). “E” is a
more than 180 degrees.
regular pentagon. “A” and “E” are convex
pentagons, while “C” is a concave non-regular
pentagon. “B” and “D” are not polygons.
Angle: An angle can be thought of in three
distinct ways: as an amount of rotation, as a
wedge shape, as a space between two
intersecting edges. For each of these ways of
thinking students should be able to identify the
vertex of the angle, and the rays that make the
sides or arms of the angle. Students use an
angle-ruler to measure angles.
Benchmark: Students use 90 degrees as a
benchmark. They have to recognize a right angle
in various orientations. Other angles such as 45
degrees (half a right angle) or 30 degrees (third
of a right angle) or 150 degrees (nearly 2 right
angles) can be estimated by comparing with a
right angle.
A square is regular; a non-square rectangle is not
(because the sides are not congruent)
A slice of pizza is a wedge shape. If we slice a
pizza into 6 equal pieces then the angle at the
center of each wedge will be 60 degrees.
A tree branch comes off the trunk of a tree at an
angle. If the angle is small then the branches
leave little space between the trunk and the
branch (like a poplar tree). If the angle is large
then the branches leave a lot of space (like a pine
tree).
This is a simplified version of an angle ruler. The
circular part is marked off as a scale, in degrees.
The vertex of the angle is at the center of the
circular part. The sides of the angle should line up
with the line segments on the angle ruler, so that
the number of degrees to rotate from one leg to
another can be counted from zero. Since the
arms of the angle can be thought of as rays (fixed
starting point, infinite length) the length of the
arms is not relevant to the size of the angle.
The above are all examples of 90 degree angles.
The orientation and the length of the legs are
irrelevant.
Angle sum of polygon: (Students first measured
and then reasoned about the angle sum of any
polygon.) By splitting any polygon into triangles
they arrive at the formula that
Angle sum = (n-2)(180 degrees), where n is the
number of sides. This works for any polygon,
regular or not.
Each angle of a regular polygon: If all angles
are equal then each angle of a regular polygon
= (n – 2)(180)/n degrees.
In the pentagon on the left we make 3 triangles by
drawing diagonals from a single vertex. The
angles of these triangles comprise the angles of
the pentagon. Thus, the angle sum of the
pentagon = 3(angle sum of triangle) = 3(180) =
540 degrees. Students can reason why there are
2 fewer triangles than sides of the original
polygon: hence, (n – 2)180.
In the pentagon on the right, similar reasoning
arrives at angle sum of pentagon = 5(angle sum
of triangle) – 360 = 540 degrees. Notice that the
central angle has to be subtracted because it is
not an angle at any of the vertices of the
pentagon.
arrives at angle sum of pentagon = 5(angle sum
of triangle) – 360 = 540 degrees. Notice that the
central angle has to be subtracted because it is
not an angle at any of the vertices of the
pentagon.
Note: this reasoning works for regular and
non-regular polygons. But if we have a regular
pentagon (for example) we can divide the sum by
5 to get the size of each angle: 540/5 = 108
degrees.
Tesselation: Tesselation means tiling the plane
with a shape or a combination of shapes, leaving
no gaps. The mention of a “plane” implies that
whatever pattern has been chosen should go on
forever.
Regular polygons that tile: Triangle,
quadrilateral, hexagon.
Non-regular polygons that tile: Any triangle or
quadrilateral tiles (some pentagons, or other
polygons also). The goal is not to memorize this
but to understand that the angle sum being a
factor of 360 degrees guarantees that the angles
of ANY triangle or quadrilateral will fit around a
vertex with no gaps.
The above is an example of tiling with a
parallelogram. Because opposite sides are
parallel and congruent the “next” parallelogram
always fits along the side of the “last”
parallelogram, creating one continuous set of
parallel lines running horizontally across the
plane, in this example.
Students find out, by experimenting, that they can
start with any triangle and make a copy that has
been rotated, as above. This always results in a
parallelogram. (At this point students observe
this. Later, when they know more about
symmetries and parallel lines they can confirm
that this will always be true.) So, since ANY
triangle can produce a parallelogram, and since
ANY parallelogram will tile a plane, we can show
that ANY triangle will tile a plane.
Notice that at the vertex highlighted there are 6
triangles coming together. The 6 angles around
this vertex are actually the angles of the original
triangle, occurring in pairs. Since the sum of the
angles of a triangle is 180 degrees, the sum of
these 6 angles is 360 degrees, which is WHY the
tiling works.
Symmetries:
A shape that can rotate around a point and still fit
inside the same outline has rotation symmetry.
A shape that can flip over a line and still fit within
the same outline has reflection symmetry. In
this unit this is used to reason informally about
certain polygons, and about tessellations. In
future units students will investigate symmetry
more thoroughly.
Conditions for a triangle: A triangle can be
made with any three lengths, as long as the sum
of any 2 sides is greater than the third side.
Given three sides that make a triangle the triangle
will be unique. It may have to be reflected or
rotated but the final shape will be congruent to the
original. The triangle is rigid.
this vertex are actually the angles of the original
triangle, occurring in pairs. Since the sum of the
angles of a triangle is 180 degrees, the sum of
these 6 angles is 360 degrees, which is WHY the
tiling works.
A parallelogram has rotation symmetry of 180
degrees around the midpoint of the diagonals:
An isosceles triangle has reflection symmetry:
An equilateral triangle has both kinds of
symmetry, reflection and rotation.
The lengths 3, 5, 5 units will make a triangle. In
fact once the triangle has been created it is
unique.
The lengths 1, 3, 5 units will NOT make a triangle;
the two sides 1 and 3 units will not add up to
enough to “close” the triangle.
Classification of triangles: A scalene triangle
has no equal sides; an isosceles triangle has 2
equal sides (and angles); an equilateral triangle
has all sides (and angles) equal.
Conditions for a quadrilateral: By using
polystrips students discovered that, given a
useable combination of 4 side lengths, many
different quadrilaterals can be created, because
the angles can change. Unlike a triangle the
quadrilateral is not rigid.
The two quadrilaterals below have the same
lengths of sides, in the same order. However the
angles are different, so the quadrilaterals are not
congruent.
Classification of quadrilaterals: Students use
squares, rectangles, parallelograms, trapezoids
and kites in their attempts to make tessellations.
Each of these is a special kind of quadrilateral. A
parallelogram has opposite sides parallel (which
permits a rotation around the intersection of the
diagonals, to fit the same outline). Thus a square
and rectangle are special kinds of parallelograms.
A rectangle has right angles, and so does a
square, but a square also has all sides equal.
Thus a square is a special kind of rectangle. A
trapezoid has only one pair of opposite sides
parallel. The goal is not to memorize the names
but to reason about the properties.
Parallel Lines and transversals: Given two
parallel lines and a transversal line crossing,
certain pairs of congruent angles are formed.
Corresponding angles are equal. Alternate
interior angles are equal.
A
C
B
D
“A” is a rectangle, “B” is a parallelogram, C is a
square, and “D” is a trapezoid.
Line 1
b
a
d
c
g
h
e
k
Line 2
f
Line 3
Line 4
If we are given that Line 1 is parallel to Line 2 then
we can conclude that a = h (corresponding
angles) and b = g (corresponding) and c = e
(corresponding) and d = h (alternate interior) and
c = g (alternate interior).
Conversely, if we know that f = k then we can
deduce that Line 3 is parallel to line 4.
Note: students do a lot more reasoning with
parallel lines in later units.